While Stealer of Souls isn't as broken in Standard as it is in Wild, it's still pretty good. However, without immunity getting too greedy can prove fatal. This deck runs relatively cheap cards and cards that provide discounts, so you can get plenty of value without your life dropping too low.

I've had a lot of fun with this deck and so far the win rate is fine as well, although I haven't played it beyond Silver yet. It is also a quick way to complete the "Ugh, It's MY Life, Mom" achievement (Spend 100 Health on Warlock cards).

Game Plan

Versus aggro, use removal and cheap minions in the early turns to not take too much damage, then summon a giant board in the mid-game to crush them.

Against slower decks, spread out your threats to exhaust your opponent's removal options. If you think the game will go very long, try to get multiple copies of Envoy Rustwix using Demonic Studies, Felosophy, Raise Dead and Kanrethad Prime.

In the long run, Soul Fragments can heal you quite lot, but as you might draw them too soon or too late, there is Man'ari Mosher as a more predictable healing option. Ideally you use them on a Void Drinker that sticks or with Nightshade Matron, but don't wait too long: healing for 4 or 6 now is better than dying with the Mosher still in your hand. Cloning a Mosher with a non-outcast Felosophy can be entirely reasonable in any matchup where your life total gets pressured.

Mulligan

Spirit Jailer and Demonic Studies are always decent keeps.

Against aggro, keep Soul Shear, Midway Maniac, Kanrethad Ebonlocke, Man'ari Mosher and possibly Fire Breather. If you already found some minions, Felosophy can be a keep as well, especially if you can get it into outcast position. Raise Dead is also useful to recycle your 1/2-drops. Void Drinker can be a keep if you're on the coin and found Spirit Jailer and/or Soul Shear already.

Void Drinker can be good against mid-range decks, if you can activate it and duplicate it using Felosophy or Raise Dead, since one is often not enough.

Against slower decks, it's worth keeping Stealer of Souls plus some card draw, or just card draw if you're not offered Stealer.

Synergies

All minions except Kanrethad Ebonlocke and some of the shuffled primes are demons. So it should be easy to get value out of Free Admission and Man'ari Mosher. Also your own boards will be mostly immune to Fire Breather. Be careful though that some opponents also play demons; I accidentally left an Illidari Inquisitor alive earlier tonight.

The discount of Kanrethad Ebonlocke also works for demons that cost life to play.

This deck is actually a rework of a demonlock deck I built earlier, which landed in the all-too-common "it's not bad, but it's just too fair" category. The addition of Stealer of Souls gives it the occasional burst of brokenness that modern Hearthstone requires. However, there are also games where Stealer of Souls plays a very limited role; there is no need to go all-in on that single card.

The hardest part is probably how much you want to draw and how much you want to play when Stealer of Souls is active. I haven't really figured that out myself yet. It does depend on the opponent of course: if your opponent needs minion or weapon damage to go face, you can go low on life to play a taunt, but if your opponent has burn spells or AoE removal, it's better to play your minions at a slower rate.

Replacements

In general, be careful with adding cards above 5 mana or non-demon minions, since you'll create anti-synergies that way. It might still be worth it if the card is good enough; I had Soulciologist Malicia in the deck at some point, for example.

Void Drinker could be replaced by Circus Amalgam. 

Drain Soul is good for board control and healing; I couldn't find a place for it, but if you have a slot to fill, it's a good pick.

C'Thun, the Shattered might be interesting to try, since you can draw through the deck pretty quickly. I don't know how often you'll have the 10 mana/life to play C'thun itself though before the game ends.